Find the greatest product of two number whose sum is 12?

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isaaq

1 week ago

1 week ago

Let x be the first number. The sum of the two numbers is 12 so 12-x is an expression for the second number.

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Let y be the product of the two numbers.

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Then an equation that represents this relationship is

y=x%2812-x%29

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Expand the right hand side of the equation.

y=12x-x%5E2

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Re-order the terms.

y=-x%5E2%2B12x

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This is a quadratic equation; its graph is a parabola. Since the leading coefficient is negative the vertex represents the maximum value of the equation.

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If the vertex of a parabola is the point (h,k), then h=-b/2a. In the formula, a is the coefficient of the x-squared terms, and b is the coefficient of the x-term when the quadratic equation is in general form [y=ax^2+bx+c].

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In our equation, a is -1 and b is 12.

h=-b%2F2a

h=%28-12%29%2F%282%2A%28-1%29%29

h=%28-12%29%2F%28-2%29

h=6

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The equation h=6 means that the x-coordinate of the vertex is 6. Substitute 6 for x in the quadratic equation to find the y-coordinate.

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y=-x%5E2%2B12x

y=-%286%29%5E2%2B12%286%29

y=-36%2B72

y=36

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The y-coordinate of the vertex is 36, so the vertex is (6,36).

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Now we need to interpret our answer in terms of this problem. Recall that x is the first number. The first number is 6. The second number is also 6 since 12-x=12-6=6. The y-coordinate is the product of the two numbers, 36.

.

Let y be the product of the two numbers.

.

Then an equation that represents this relationship is

y=x%2812-x%29

.

Expand the right hand side of the equation.

y=12x-x%5E2

.

Re-order the terms.

y=-x%5E2%2B12x

.

This is a quadratic equation; its graph is a parabola. Since the leading coefficient is negative the vertex represents the maximum value of the equation.

.

If the vertex of a parabola is the point (h,k), then h=-b/2a. In the formula, a is the coefficient of the x-squared terms, and b is the coefficient of the x-term when the quadratic equation is in general form [y=ax^2+bx+c].

.

In our equation, a is -1 and b is 12.

h=-b%2F2a

h=%28-12%29%2F%282%2A%28-1%29%29

h=%28-12%29%2F%28-2%29

h=6

.

The equation h=6 means that the x-coordinate of the vertex is 6. Substitute 6 for x in the quadratic equation to find the y-coordinate.

.

y=-x%5E2%2B12x

y=-%286%29%5E2%2B12%286%29

y=-36%2B72

y=36

.

The y-coordinate of the vertex is 36, so the vertex is (6,36).

.

Now we need to interpret our answer in terms of this problem. Recall that x is the first number. The first number is 6. The second number is also 6 since 12-x=12-6=6. The y-coordinate is the product of the two numbers, 36.

1 week ago

the other =12-x

the product be x

x= x(12-x)

x=12x - x^2

dy/dx=12 - 2x

at maximum height dt/dx=0

12 - 2x=0

-2x=-12

x=-12/-2

x=6

d^2y/dx^2= -2< 0

since x=6 the maximum value of y

=6(12-6)

y=36